Why do higher frequencies tend to have a smaller amplitude than low frequencies?

Question

Complex waves found in nature tend to have downward sloping frequency distribution. This isn't to say the fundamental always has the highest level but the general trend is that higher overtones tend to have lower levels.

The simple answer is that energy is proportional to the square of frequency and square of amplitude and so increasing frequency must mean amplitude decreases, however that's assuming the energy of each mode is the same.

Ok so what about the equipartition theorem? When a system has multiple degrees of freedom, over time the energy will become evenly distributed among the modes. But this assumes the modes are coupled in a significant way. If we're talking about a superposition of sines like a plucked guitar string there should be very little coupling (an idealized string should have 0 coupling since the harmonics are orthogonal).

However taken from here

Whether the modes are orthogonal or not really only covers the question of whether they interact via linear processes, e.g. superposition. Orthogonal modes can still interact with each other nonlinearly, which is typically the case when one observes harmonics (assuming they are real harmonics and not spontaneously-excited modes that happen to be at twice the frequency of another mode). This most commonly takes the form of quadratic phase coupling, i.e. there are quadratic terms in the dependent variable in the governing equations. That allows energy exchange between otherwise normal modes. While a guitar strings is likely pretty close to ideal for small amplitudes, it would be fairly unusual for a real-world problem with all the aforementioned complications to not include some degree of nonlinearity at higher amplitudes.

Then there's the purely mathematical answer: the total energy of the vibration has to be finite. If we have an infinite number of possible modes of vibration you need some distribution of the energy between few of them and you get less and less energy left for higher ones (a la convergence).

Now this isn't just for instruments, it also happens with electrical signals. I found this answer regarding electrical signals

the high frequency components in signals are by passed by the capacitive and inductive effects of the line. The value of these reactive components is such that they alter the higher frequencies more than the lower ones.

Note: I'm not saying the that the lowest frequency has to be the loudest followed by the second harmonic, etc. A trumpet is a clear counter example to this. Just that the general trend is that the lower frequencies in a sound tend to dominate and the higher harmonics tend to decrease in amplitude the higher up you go.

In my searching I stumbled upon this

A general answer is that most physical systems are low pass filters. They attenuate high frequencies more than low frequencies. There are exceptions, but most things are like that.

This makes sense for attenuation since attenuation is determined by the inverse of frequency but I'm talking about why higher frequencies aren't even excited to any significant degree in the first place

Edit: I also came across this: "Weighting harmonics proportionally to the square of the frequency is equivalent to differentiating twice and so gives a measure of the reciprocal of the radius of curvature of the wave-form and is therefore related to the sharpness of any corners on it"

of course the frequency spectrum of a wave is related to the shape of the wave but I think this might play a part in why we tend to not observe waves where the high frequencies have a high amplitude.

Answer 1

There's a number of oscillatory phenomena, with different physical character, that follow this kind of rule.

Electric waves (light, radio, X-rays etc.) are capable of interacting with matter in such a way as to downshift the frequency; an X-ray hitting a fluorescent screen is a classic example. This is because the energy per photon of the X-rays can create many low-energy visible light photons, but cannot be stored and combined with more X-rays, or otherwise increased, so as to allow a frequency upshift. Higher frequency photons have more energy than lower frequency, so conservation of energy prohibits their production.

Electric signals in wires are strongly attenuated at high frequencies because of wire resistance and skin effect (but that's negligible in your house wiring at 60 Hz), and because of dielectric absorption (the solid insulating materials that we use in wiring are less than perfectly 'elastic' and steal small amounts of electric energy on each cycle; more cycles per second leads to higher losses).

Sonic energy is also diminished according to the elastic imperfections in materials carrying the sound, so(for instance) a bell may have four or five harmonics in its first millisecond after striking, but the resonating sound after a second is almost entirely the lowest harmonic of those frequencies. This is because the resonator has a Q value, roughly diminishing the mechanical energy by a percentage on each cycle. More cycles per second (higher frequency) means less persistence of the higher frequencies in the sound.

Finally, all wave phenomena carrying energy are subject to the laws of thermodynamics; energy will convert from any distribution down to the thermal distribution of its surroundings, given enough interactions. Thus, the 'background noise' we hear, is eventually going to be a thermal distribution, and the Boltzmann law says this trails off at high frequency for light, and gives comparable predictions for other quantized energies.